AN ANALYSIS OF LEARNERS’ SOLUTION STRATEGIES IN THE

ISSN 2087-8885 E-ISSN 2407-0610 Journal on Mathematics Education Volume 11, No. 3, September 2020, pp. 501-512

501

AN ANALYSIS OF LEARNERS’ SOLUTION STRATEGIES IN THE CONTEXT OF MODELLING TASKS

Xenia-Rosemarie Reit1, Marc Schäfer2 1University of Duisburg-Essen, Thea-Leymann-Straße 9, 45127 Essen, Germany

2Rhodes University, PO Box 94, Makhanda, (Grahamstown) 6140, Eastern Cape, South Africa Email: [email protected]

Abstract

It remains a challenge for teachers to integrate modeling tasks in everyday mathematics classes. Many studies have been conducted that show the difficulties faced by teachers. One of the challenging aspects in this regard is that of assessment. In the present study, a connection between structures of learners’ solution strategies and cognitive considerations is established to develop a practice-oriented instrument to determine and assess the complexity of solution strategies of modeling tasks. In this paper, the selected learners’ strategies’ structure was analyzed in-depth to identify the underlying cognitive structure. The results show that thought operations carried out in parallel complicated a solution strategy. However, the results also support a purely sequential thought operation approach without weighting parallel thought operations, which corresponds to an intuitive assessment procedure by mathematics teachers. As assessment is a great challenge for many teachers in the context of modeling tasks, this study provides a promising frame of reference for further research in this important domain of assessment and modeling.

Keywords: modelling tasks, cognitive structure, solution strategies, mathematics education

Abstrak Ini merupakan suatu tantangan bagi guru untuk mengintegrasikan tugas pemodelan dalam kelas matematika sehari-hari. Banyak penelitian telah dilakukan yang menunjukkan kesulitan yang dihadapi oleh para guru. Salah satu aspek yang menantang dalam hal ini adalah penilaian atau evaluasi. Dalam penelitian ini, hubungan antara struktur strategi penyelesaian peserta didik dan kemampuan kognitif dibangun untuk mengembangkan instrumen berorientasi praktik untuk menentukan dan menilai kompleksitas strategi penyelesaian tugas pemodelan. Makalah ini membahas tentang struktur strategi peserta didik yang dipilih, dianalisis secara mendalam untuk mengidentifikasi struktur kognitif yang mendasarinya. Hasilnya menunjukkan bahwa operasi pemikiran yang dilakukan secara paralel mempersulit strategi penyelesaian. Namun, hasil ini juga mendukung pendekatan operasi pemikiran sekuensial murni tanpa pembobotan operasi pemikiran paralel, yang sesuai dengan prosedur penilaian intuitif oleh guru matematika. Hal ini dikarenakan penilaian atau evaluasi merupakan tantangan terbesar bagi banyak guru dalam konteks tugas pemodelan, penelitian ini memberikan kerangka acuan yang menjanjikan untuk penelitian lebih lanjut dalam domain penting penilaian dan pemodelan ini.

Kata kunci: soal-soal pemodelan, struktur kognitif, strategi-strategi penyelesaian, pendidikan matematika

How to Cite: Reit, X-R., & Schäfer, M. (2020). An Analysis of Learners’ Solution Strategies in the Context of Modelling Tasks. Journal on Mathematics Education, 11(3), 501-512. http://doi.org/10.22342/jme.11.3.11345.501-512.

This paper addresses the observation that mathematical modelling tasks generally do not feature

prominently in everyday school life. As teachers are directly responsible for the implementation of

educational recommendations and standards, such as incorporating modelling tasks in the teaching of

mathematics, they are the key agents of change in ensuring that this happens (Fernandes, 1995; Wilson

& Cooney, 2002). Mathematical modelling is cognitively demanding, for both students and teachers

(Blum & Ferri, 2009) and implementing modelling in school is thus a great challenge for both students

and teachers (English, 2009; Rivera, 2018; Warwick, 2007). Besides the organizational and material-

oriented challenges inherent in using modelling tasks, other problems teachers face relate to the open-

502 Journal on Mathematics Education, Volume 11, No. 3, September 2020, pp. 501-512

ended nature of these modelling tasks. Open-ended tasks lack predictability and make use of a variety

of solution strategies, thus complicating assessment (Schmidt, 2010). This requires teachers to adopt a

pedagogy that reflects this openness to working with different solution strategies.

This paper reports on a bigger research study which specifically focused on a particular method

to determine the complexity of solution strategies of modelling tasks. It is hoped that the findings of

this research might lead to easier manageability of modelling tasks and can also be a starting point for

alternative ways of assessment in the teaching of modelling tasks. Therefore, the study considered the

following main question: does a thought structure analysis, together with its underlying cognitive and

structural assumptions, lead to a suitable characterization of the complexity of solution strategies to

modelling tasks?

When assessing the solution to mathematics tasks, their numerical value is often taken as the

dominant indicator of a right or wrong solution. When dealing with modelling tasks, however, it is often

not meaningful to merely assess a solution’s numerical value due to the multiple solution nature of these

modelling tasks. A common observation is that modelling tasks cannot be assessed as objectively as

traditional task formats (Spandaw & Zwaneveld, 2010). Studies show that assessment of modelling

tasks is a major challenge, which often hinders mathematics teachers from integrating modelling in their

mathematics classes (Jensen, 2007). The complexity of the solution is a critical measure when assessing

student solutions (Shanta & Wells, 2020). Therefore, if we want modelling to be part of mathematics

teaching, ways of practical assessment need to be found. This study was thus undertaken to determine

the complexity of solution strategies to selected modelling tasks, which can then form the starting point

for further assessment considerations.

The Role of Cognitive Structures

A common procedure when solving mathematical problems in general is what Bourbaki (1961,

p.163) refers to as “recalling structures”. In school mathematics learners are often taught to structure a

solution in terms of “a question”, “calculation” and “the answer”. Those structuring procedures are

meant to relieve and support the learners’ thinking processes and help to objectivize the individual

solution strategies. The latter is closely connected to assessment practices where structures also play an

important role. A common assessment procedure that mathematics teachers often use is to provide a

sample solution in advance and to identify the necessary intermediate steps to attain a solution which is

then scored. Because solutions to modelling tasks are often open-ended and can vary widely, there is,

however, no unique sample solution. This complicates an objective assessment of solutions.

The common, mostly intuitive, assessment procedure mentioned above has similarities with

structural considerations (e.g. the use of arithmetic trees, Figure 1) within the field of word problems.

Breidenbach (1969) looked at the structural-substantial complexity of a word problem among other

things, to determine its complexity. For example, in the task illustrated in Figure 1 there are three

variables in the first step, where the third variable is uniquely determined from the other two by one

Reit & Schäfer, An Analysis of Learners’ Solution Strategies in the Context of Modelling Tasks 503

operation (in this case it is multiplication). Breidenbach called this arrangement “Simplex”. A linking

of several Simplexes in the solution process is called a “Komplex”. Further developments by Winter

and Ziegler (1969) lead to the arithmetic tree illustrated in Figure 1, which is still used in mathematics

textbooks today.

An obvious, but so far empirically not validated conclusion, is that a large number of Simplexes

and a complicated nesting of them, have an effect on the complexity of the tasks’ solutions. Cohors-

Fresenborg, Sjuts, and Sommer (2004) describe task complexity, inter alia, by the criterion “cognitive

complexity” of thought processes. This allows for simultaneity or nesting of what we call “thought

steps”. In this context Kaune (2000) also distinguishes between two aspects of cognitive complexity.

These are firstly the processing of multiple information within one thought step, and secondly the

influence of previous information and data.

Figure 1. An example of an arithmetic tree for a selected task according to Winter and Ziegler (1969)

Within the modelling discourse, cognition and assessment of thinking has particular relevance to

study how students deal with modelling tasks. Blum and Ferri referred to students’ difficulties with

modelling tasks by the tasks’ cognitive demand (Blum & Ferri, 2009, p. 46). Tanner and Jones

investigated the influence of (meta)cognitive skills on modelling (Tanner & Jones, 1993; 2002) and

assessed, inter alia, cognitive ability.

Cognitive Complexity

The present study investigates the cognitive complexity of a solution strategy by analyzing the

structural complexity represented by its arithmetic tree-like structure as illustrated in Figure 1. The

coherence of structural considerations and cognitive theories play an important role in solving

modelling tasks. In their study, Fletcher and Bloom (1988) assumed that text comprehension is an

important problem-solving process in itself, where the problem solver must find a causal chain to link

the start to the end of the text. In order to form these links, Fletcher and Bloom (1988) posited that the


information concerning the task must be kept in the mind of the problem solver. Results of their study

show that it is important for the problem solver to access the immediately preceding information before

proceeding to the next step in the causal chain. It can be concluded that the role of the working memory

is to keep key information available. This is necessary to link old and new information (Baumann,

2000). By relating these findings to structural considerations of a particular solution strategy represented

as an arithmetic tree, statements can be made about its cognitive complexity. On the one hand, an

arithmetic tree-like structure (Figure 1) can be interpreted as a causal chain since the start (the given

information in the task text) and end (the solution of the task) is linked by chain links. In the context

of solving mathematics tasks we refer to these links as intermediate steps in the solution process. On

the other hand, the immediately preceding information can be identified as relevant intermediate steps.

Thus, it can be concluded that the immediately preceding intermediate step must be kept active in the

working memory to master the next step in the chain. The assumption that mental processing capacity

is limited (Sweller, 1988), leads to the conclusion that a plethora of information which has to be kept

active at the same time, can complicate the solution process. Thus, it can be concluded that the working

memory load is dependent on the number of intermediate steps necessary to master the current

intermediate step in the problem-solving process. This means that the load of the working memory

increases with the increasing amount of information needed at various points in the solution process.

Based on these theoretical considerations the following definition of thought structure was taken

fundamental for the present study (Reit, 2016, p. 58):

“A thought operation is a necessary structuring process for finding a solution, which results directly (i.e. without intermediate calculations) from one or several previous data. Previous data are either (intermediate) results from previous thought operations or given data from the tasks’ text.”

A thought structure thus illustrates the logical sequence of thought operations (Reit, 2016, p. 60).

The present study establishes a framework to structurally analyze student solutions to modelling tasks

with the aim of determining their complexity. It is assumed that the number of sequential and parallel

thought operations provides information about the cognitive complexity of a solution strategy.

METHOD

This study is framed by three main phases (see Figure 2 for phase one and two). In phase one an

empirical study was conducted with 1800 student solutions to selected modelling tasks. The different

solution strategies were identified. In phase two of the study, a so-called thought structure analysis

(Reit, 2016) was conducted. Here the structures of the various solution strategies were analyzed by

identifying their specific thought operations. At this point the number of parallel and sequential thought

operations were identified. To translate this data quantitatively, statistical models were developed.

Some of these models differentiated between sequential and parallel thought operations by applying a


weighting to the latter. Of particular interest were the differences in the models that applied or did not

apply a weighting to parallel thought operations, as we wanted to determine which model could most

appropriately identify the complexity of the solution strategies of modelling tasks. In a third phase the

evaluation of the complexity models took place. The theoretical complexity of the solution strategies

was statistically compared to the average score of the student solutions as a measure for the empirical

complexity. This was based on a rubric. In particular, we analyzed which complexity model led to a

statistically verifiable and substantially reasonable coherence between complexity and score in terms

of “high score, low complexity” and vice versa. These results finally yielded information about the

influence of parallel thought operations on the complexity of the solution strategies.

Figure 2. Method in phase 1 and phase 2 of the study

For the bigger study referred to above, five modelling tasks were developed according to

predefined criteria (see Reit & Ludwig, 2015a; 2015b). Three of these five tasks were randomly selected

and published in a booklet which was distributed to the 1800 Grade 9 students (15 years of age) from

German grammar schools. Each student was asked to solve the three tasks individually. Each student

had 45 minutes to complete the booklet. For the purpose of this paper we will discuss “the Potato

problem” as illustrated in Figure 3. The students were asked to mathematically estimate the number of

French fries which can be cut out of a single potato.

Figure 3. The potato modelling task


Thought Structure Analysis

In phase one of the bigger study (Figure 2) all student solutions were clustered into different

solution strategies according to which mathematical model was used in the solution process (for a

detailed discussion of the different solution strategies see Reit, 2016) (interrater reliability: Cohen’s

κ∈[0.702-0.949]). In a second step, each solution strategy was analyzed to reveal its thought structure

- see Figure 4 for the Potato modelling task. In particular, single thought operations were identified and

a thought structure was set up. This provided information about the number of parallel and sequential

thought operations that took place within the respective solution strategy.

Figure 4 illustrates a particular student solution to the potato modelling task, representing the

solution strategy layer and its corresponding thought structure. The potato and the French fry were

approximated to and represented by cuboids. By calculating the number of French fries fitting into

height and width of the potato and multiplying these results, the total number of French fries could be

mathematically estimated (Figure 4). The thought structure illustrates the thought operations needed for

the next step in the solution process. This can be represented as a vector representation ((3,1,1) for the

solution strategy in Figure 4), indicating the number of parallel thought operations in each thought

structure level. So, in this case there are 3 thought structure levels which have to be mastered

consecutively to come to a final solution. In the first level there are 3 thought operations to be linked

together mathematically correctly to come to the second level with one thought operation, and in the

third level there is also one thought operation to be mastered.

Figure 4. Student solution (solution strategy layer) (right) together with its thought structure (left),

indicating the number of parallel thought operations per level

It is recognized that besides the cognitive aspects, linguistic formulations also play a role when

characterizing the complexity of a modelling task (e.g. Cohors-Fresenborg, Sjuts, & Sommer, 2004;

Walzebug, 2014, p. 161). Since these aspects are not considered by the thought structure model

explained above, the complexity of the task text is integrated as a separate factor (see Reit 2016, pp.


103). A scheme following Cohors-Fresenborg et al. (2004) was developed by Reit (2016, p. 104) which

classifies the complexity of the tasks’ text and adds to the cognitive complexity of a student solution.

Complexity Models

The empirical complexity represents the average (normalized) score of the student solutions

being clustered to the respective solution strategy. The thought structures currently provide a vectorial

representation which we subsequently translated into a scalar. The question arises whether parallel

thought operations complicate a solution strategy. Several models (Table 1) were used to translate the

vectorial representation of the number of parallel and sequential thought operations into a scalar value.

They differ in the kind of weighting of the parallel thought operations. The aim of determining a

theoretical complexity was not primarily to find the best possible fit of a possible complex complexity

model, but rather to find a model that could be used in school practice, and that, at the same time

reflected the data well.

Table 1. Different Complexity Models Applied to the Potato Modelling Task

The addition model in Table 1 simply adds up the number of thought operations which results in

a theoretical complexity of 5 for the solution in Figure 4. It does not differentiate between parallel and

sequential thought operations. This, we argue, is possibly the most widespread assessment procedure

of mathematics teachers: scoring of intermediate steps within a solution and subsequent addition of

these partial scores. The results of the addition model then show whether this so far intuitive assessment

procedure can be empirically supported and thus transferred also to modelling solutions. The maximum

model refers to the maximum metric of a vector. Using this model, the theoretical complexity is

determined by the thought structure level with the highest number of parallel thought operations.

An obvious way to translate a vector into a scalar is to calculate its norm. This is referred to as

the norm model. This leads to a quadratic weighting of parallel thought operations as is also done by

the quadratic model. Another possibility of operationalizing the theoretical complexity is based on the

consideration that the sequence of processing thought operations on the same level in the thought

structure is arbitrary. This content-related notion is reflected by the factorial model. Following the

factorial model, parallel thought operations are seen to be cognitively more demanding and thus result

in a more difficult solution strategy than sequential ones.

Thought Structure Vector (3,1,1) Theoretical Complexity

Addition model 3+1+1 = 5

Maximum model max(3,1,1) = 3

Norm model √32 + 12 + 12 ≈ 3.3

Quadratic model 32 + 12 + 12 = 11

Factorial model 3!+1!+1! = 8


This theoretical complexity was then compared to an empirical complexity representing the

average (normalized) score of the student solutions being clustered to the respective solution strategy.

For this purpose, a rubric was set up based on the thought structures of the solution strategies so that

each thought operation corresponded to an essential intermediate step of the solution. Identifying and

assessing essential intermediate steps and scoring them corresponds, we argue, to the common practice

for evaluating written student solutions in mathematics. Depending on whether a thought operation

was executed completely correctly, with errors, or not at all, 1, 0.5 or 0 credits were awarded. Follow-

up errors were considered, as calculation errors should not be excessively weighted.

RESULTS AND DISCUSSIONS

In this section a comparison is drawn between theoretical difficulty, resulting from the thought

structure analysis and operationalized by different complexity models, and the average score of a

solution strategy as a measure for its empirical complexity. At first, solution strategies are considered

and compared to each other. Then the method is extended to complete modelling tasks where both

theoretical and empirical difficulties of the tasks’ solution strategies are averaged to characterize the

task complexity. To compare theoretical and empirical complexity, a power regression is implemented

with a calculation of its goodness of fit in terms of a pseudo-R²-value. This allows for a statistical

evaluation of the substantially reasonable relationship that “the more difficult a solution strategy is, the

lower is the respective score”. Furthermore, this supports a substantial conclusion with regard to the

influence of parallelism of thought operations on the complexity of a solution strategy.

Complexity of Solution Strategies

A comparison of the theoretical and empirical complexity depending on the respective

complexity model (see Table 1) shows that the factorial model leads to the best results concerning the

goodness of fit of the regression model (Figure 5). This result speaks to the suitability of the factorial

model for the characterization of the theoretical complexity of solution strategies. The factorial model

weights parallel thought operations which leads to the conclusion that parallelism has an effect on the

complexity of a solution strategy. The maximum model with a pseudo-R²-value of only 0.15 turns out

to be inadequate. The addition model with a pseudo-R²-value of 0.69 however, proves to be adequate.

The norm and quadratic model show weak pseudo-R²-values of 0.52, which suggests only a limited

suitability.

These results suggest that parallelism of thought operations has an influence on the complexity

of the solution strategy, although the kind of weighting used in the analysis of these solution strategies

seems to be crucial. This is also underlined by the fact that the application of the addition model can

lead to better results than most of the weighting models (maximum, norm and quadratic model) which

nevertheless also indicates that a pure addition of thought operations might also be a reasonable analysis

strategy.


Figure 5. Comparison of the average score as a measure for the empirical complexity and theoretical

complexity of solution strategies by a power regression and its goodness of fit in terms of a pseudo-

R²-value

The main question this paper asks is whether the thought structure analysis together with its

underlying cognitive and structural assumptions, leads to a reasonable and suitable characterization of

the complexity of solution strategies of modelling tasks. The results presented above suggest that the

factorial model is a suitable analytical tool to analyze the complexities of solution strategies when

solving modelling tasks. The factorial model capitalizes on, and considers, the complexities associated

with parallel thought operations, which aligns well with Sweller’s cognitive load theory (Sweller, 1988).

The good performance of the factorial model confirms that parallelism of thought operations has an

influence on the complexity of solution strategies.

Nevertheless, it should also be emphasized that that addition model, as a non-weighting model

leads to statistically good results. This suggests that an over-reliance of weighted complexity models


and the influence of parallelism of thought operations on complexity cannot be assumed. Furthermore,

the statistically good results of the addition model also support the so far intuitive procedure of many

mathematics teachers’ assessment of mathematics tasks. A common practice among teachers’

assessment of learners’ solution strategies is to identify reasonable intermediate steps in a solution

which are worthwhile scoring. Assuming that the assessment of solutions to a mathematics task is

strongly determined by the complexity of the task, the results of the addition model supports the so far

intuitive assessment practice, and we suggest that this practice can also be applied to modelling tasks.

Baird, Hopfenbeck, Newton, Stobart and Steen-Utheim (2014, p. 21) state that “assessments

define what counts as valuable learning and assign credit accordingly”. It is also supported by Niss’

proverb that “what you assess is what you get” (Niss, 1993).

CONCLUSION

Assessment is of great importance in prescribing and guiding what happens in a mathematics

class. Notwithstanding that assessment is a great challenge for many teachers in the context of

modelling tasks, this study provides a promising frame of reference for further research in this important

domain of assessment and modelling.

ACKNOWLEDGMENTS

This work was funded by Stiftung Polytechnische Gesellschaft Frankfurt. Thanks to Jean Schäfer

for proofreading the paper.

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AN ANALYSIS OF LEARNERS’ SOLUTION STRATEGIES IN THE